Deciphering Gödel’s Proof: Exploring the Genius Behind Gödel’s Theorem and its Impact on Mathematics” In 1931, the Austrian philosopher Kurt Godel pulled off arguably one of the greatest intellectual achievements of the past.
Mathematicians of the time searched for a solid foundation to mathematics. They sought a set of fundamental mathematical concepts, also known as axioms, which were solid and never leading to conflicts that served as the basic the blocks of all mathematical truths.
However, his shocking incompleteness theorems, which he published at the age of 25 years old, destroyed that idea. He demonstrated the fact that every set of axioms that you might consider as a potential basis for maths will eventually be insufficient;
There are always truths regarding numbers that are not established by these axioms. He also demonstrated that no possible set of axioms will be able to prove its own reliability.
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His incompletion theorems suggested that there is no mathematical explanation of everything, and no concordance between what’s proven and what’s real. The proof mathematicians have is based on their initial assumptions and not on any fundamental truth upon the which all answers stem.
For the past 89 years that have passed since the discovery of Godel, mathematical scientists have run on exactly the types of questions that his theories predicted. For instance, Godel himself helped establish his continuous hypothesis that is about the size of infinity, remains unsolved as is the halting question that inquires whether computers that are fed an undetermined input will continue to be in continuous motion or will eventually stop.
Undecidable issues have even been raised in Physics which suggests that Godelian incompleteness isn’t just a problem of math, but in a misunderstood way, reality too.
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Here’s a brief outline of the way Godel proved his theories.
Godel Numbering
Godel’s principal goal was to map assertions concerning a set of axioms on statements in the system. That is, onto statements regarding numbers. This mapping enables a system of axioms to speak clearly about the system itself.
The initial step of this process is mapping any mathematical expression or set of statements, into an uniqe number, also known as Godel. Godel number.
A slightly altered version of Godel’s plan, as that was presented in the work of Ernest Nagel and James Newman in their book of 1958, Godel’s proof It begins with 12 fundamental symbols that act as a vocabulary to express the basic axioms.
For instance, the fact that something exists could be represented by a symbol and the expression of addition is +. In addition, the symbol s which means “successor of” provides a means to specify numbers. ss0 is an example. is a reference to the number 2.
The twelve symbols are then assigned the Godel numbers 1-12.
Then, letters representing variables, beginning with the letters x, y and Z translate into prime numbers that are greater that 12 (that is 13 17 19, …).
Any combination of these variables and symbols–that is, any equation or series of formulas could be constructed will have its own Godel number.
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